Lesson 11

The purpose of this warm-up is for students to notice that figures composed of multiple arrays can be decomposed into smaller arrays, and that this is a strategy to determine the total number of dots. This will be helpful in later lessons when students decompose figures into rectangles to find the total area.

When students find ways to decompose the given arrangements of dots to find the number of dots, they practice looking for and making use of structure (MP7).

• Groups of 2
• “How many do you see? How do you see them?”
• Flash the image.
• 30 seconds: quiet think time

• Display the image.
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
• Repeat for each image.

• “What’s the same in how you found the number of dots in each image?” (I looked for groups of 4. I decomposed each arrangement of dots into smaller arrays.)
• Consider asking:
  • “Who can restate the way ___ saw the dots in different words?”
  • “Did anyone see the dots the same way but would explain it differently?”
  • “Does anyone want to add an observation to the way ____ saw the dots?”

The purpose of this activity is for students to find missing products in the multiplication table as they consider the rectangular structure of how products are organized in the table.

• Groups of 2
• Display the four blank 5-by-5 multiplication tables, each showing a multiple of 3 in the third row, in order from 3 to 12.
• “What do you notice? What do you wonder?” (Students may notice: There are numbers along the top and down the left column. In each table the number in the rectangle is going up by 3. The number in the rectangle lines up with two factors that you would multiply to get that number. Students may wonder: Why are there numbers in each rectangle? Why are there numbers along the top and on the left?)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses

• “Create your own rectangles in these tables and record the product in each table that represents the rectangle.”
• 3–5 minutes: independent work time
• “Share your rectangles with your partner and explain your reasoning.”
• 2–3 minutes: partner discussion
• Monitor for students who:
  • Create a rectangle by lining up the bottom and right edges with the corresponding factors.
  • Count the total number of squares to make sure it matches the product they record in the rectangle.
  • Record the product that represents the rectangle in the bottom right square of the rectangle.
  • Record different rectangles that both have an area of 24 square units.

• Have 3–5 previously selected students share their rectangles and how they created them.
• “This is called the multiplication table. The numbers along the top and on the left side are the factors, or the numbers we are multiplying.”
• “The number in the lower right square of the rectangle is the product. The product matches up with a factor above it and a factor to the left of it.”
• “We could represent the factors and the product in the last table in the first group of tables with 3 times 4 equals 12.”
• Display \(3 \times 4 = 12\)
• “How is the multiplication table different than other ways we’ve represented multiplication?” (It’s harder to see the factors. There’s no equal sign. We can picture a rectangle that helps you find the product.)

The purpose of this activity is for students to find products in the multiplication table. Students are encouraged to find familiar products before working on less-familiar ones. They do not need to fill in all of the products in the table. The synthesis focuses on patterns students find in the table and how they can show the patterns with equations.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?” (Students may notice: Each product is in 2 places in the multiplication table. The number 4 only shows up in one place on the multiplication table. If I go down from the top 2 or across from the 2 on the left the products count by 2. The factors count by 1 as the products count by 2, like 1 and 2, 2 and 4, 3 and 6. All the numbers in the table are even. The highest product is 10. Students may wonder: Could we put other products in the table?)
• 1 minute: quiet think time
• Share responses.

• “Work independently on the first problem.”
• 3–5 minutes: independent work time
• Share responses.
• “Let’s finish filling in the rows and columns that show products of twos and products of fives.”
• Display table showing all the products of 2 and 5.
• “Think about how you would answer the last two problems.”
• 1 minute: quiet think time
• “Now work with your partner to complete the last two problems.”
• 5–7 minutes: partner work

• “What patterns did you find in the products of 5? How did you show them with equations?”
• “What other patterns do we see in the table? How could we show the patterns with equations?”